1/*
2 * Copyright 2012 Google Inc.
3 *
4 * Use of this source code is governed by a BSD-style license that can be
5 * found in the LICENSE file.
6 */
7#include "SkLineParameters.h"
8#include "SkPathOpsCubic.h"
9#include "SkPathOpsLine.h"
10#include "SkPathOpsQuad.h"
11#include "SkPathOpsRect.h"
12
13const int SkDCubic::gPrecisionUnit = 256;  // FIXME: test different values in test framework
14
15// FIXME: cache keep the bounds and/or precision with the caller?
16double SkDCubic::calcPrecision() const {
17    SkDRect dRect;
18    dRect.setBounds(*this);  // OPTIMIZATION: just use setRawBounds ?
19    double width = dRect.fRight - dRect.fLeft;
20    double height = dRect.fBottom - dRect.fTop;
21    return (width > height ? width : height) / gPrecisionUnit;
22}
23
24bool SkDCubic::clockwise() const {
25    double sum = (fPts[0].fX - fPts[3].fX) * (fPts[0].fY + fPts[3].fY);
26    for (int idx = 0; idx < 3; ++idx) {
27        sum += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
28    }
29    return sum <= 0;
30}
31
32void SkDCubic::Coefficients(const double* src, double* A, double* B, double* C, double* D) {
33    *A = src[6];  // d
34    *B = src[4] * 3;  // 3*c
35    *C = src[2] * 3;  // 3*b
36    *D = src[0];  // a
37    *A -= *D - *C + *B;     // A =   -a + 3*b - 3*c + d
38    *B += 3 * *D - 2 * *C;  // B =  3*a - 6*b + 3*c
39    *C -= 3 * *D;           // C = -3*a + 3*b
40}
41
42bool SkDCubic::controlsContainedByEnds() const {
43    SkDVector startTan = fPts[1] - fPts[0];
44    if (startTan.fX == 0 && startTan.fY == 0) {
45        startTan = fPts[2] - fPts[0];
46    }
47    SkDVector endTan = fPts[2] - fPts[3];
48    if (endTan.fX == 0 && endTan.fY == 0) {
49        endTan = fPts[1] - fPts[3];
50    }
51    if (startTan.dot(endTan) >= 0) {
52        return false;
53    }
54    SkDLine startEdge = {{fPts[0], fPts[0]}};
55    startEdge[1].fX -= startTan.fY;
56    startEdge[1].fY += startTan.fX;
57    SkDLine endEdge = {{fPts[3], fPts[3]}};
58    endEdge[1].fX -= endTan.fY;
59    endEdge[1].fY += endTan.fX;
60    double leftStart1 = startEdge.isLeft(fPts[1]);
61    if (leftStart1 * startEdge.isLeft(fPts[2]) < 0) {
62        return false;
63    }
64    double leftEnd1 = endEdge.isLeft(fPts[1]);
65    if (leftEnd1 * endEdge.isLeft(fPts[2]) < 0) {
66        return false;
67    }
68    return leftStart1 * leftEnd1 >= 0;
69}
70
71bool SkDCubic::endsAreExtremaInXOrY() const {
72    return (between(fPts[0].fX, fPts[1].fX, fPts[3].fX)
73            && between(fPts[0].fX, fPts[2].fX, fPts[3].fX))
74            || (between(fPts[0].fY, fPts[1].fY, fPts[3].fY)
75            && between(fPts[0].fY, fPts[2].fY, fPts[3].fY));
76}
77
78bool SkDCubic::isLinear(int startIndex, int endIndex) const {
79    SkLineParameters lineParameters;
80    lineParameters.cubicEndPoints(*this, startIndex, endIndex);
81    // FIXME: maybe it's possible to avoid this and compare non-normalized
82    lineParameters.normalize();
83    double distance = lineParameters.controlPtDistance(*this, 1);
84    if (!approximately_zero(distance)) {
85        return false;
86    }
87    distance = lineParameters.controlPtDistance(*this, 2);
88    return approximately_zero(distance);
89}
90
91bool SkDCubic::monotonicInY() const {
92    return between(fPts[0].fY, fPts[1].fY, fPts[3].fY)
93            && between(fPts[0].fY, fPts[2].fY, fPts[3].fY);
94}
95
96bool SkDCubic::serpentine() const {
97    if (!controlsContainedByEnds()) {
98        return false;
99    }
100    double wiggle = (fPts[0].fX - fPts[2].fX) * (fPts[0].fY + fPts[2].fY);
101    for (int idx = 0; idx < 2; ++idx) {
102        wiggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
103    }
104    double waggle = (fPts[1].fX - fPts[3].fX) * (fPts[1].fY + fPts[3].fY);
105    for (int idx = 1; idx < 3; ++idx) {
106        waggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
107    }
108    return wiggle * waggle < 0;
109}
110
111// cubic roots
112
113static const double PI = 3.141592653589793;
114
115// from SkGeometry.cpp (and Numeric Solutions, 5.6)
116int SkDCubic::RootsValidT(double A, double B, double C, double D, double t[3]) {
117    double s[3];
118    int realRoots = RootsReal(A, B, C, D, s);
119    int foundRoots = SkDQuad::AddValidTs(s, realRoots, t);
120    return foundRoots;
121}
122
123int SkDCubic::RootsReal(double A, double B, double C, double D, double s[3]) {
124#ifdef SK_DEBUG
125    // create a string mathematica understands
126    // GDB set print repe 15 # if repeated digits is a bother
127    //     set print elements 400 # if line doesn't fit
128    char str[1024];
129    sk_bzero(str, sizeof(str));
130    SK_SNPRINTF(str, sizeof(str), "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
131            A, B, C, D);
132    mathematica_ize(str, sizeof(str));
133#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
134    SkDebugf("%s\n", str);
135#endif
136#endif
137    if (approximately_zero(A)
138            && approximately_zero_when_compared_to(A, B)
139            && approximately_zero_when_compared_to(A, C)
140            && approximately_zero_when_compared_to(A, D)) {  // we're just a quadratic
141        return SkDQuad::RootsReal(B, C, D, s);
142    }
143    if (approximately_zero_when_compared_to(D, A)
144            && approximately_zero_when_compared_to(D, B)
145            && approximately_zero_when_compared_to(D, C)) {  // 0 is one root
146        int num = SkDQuad::RootsReal(A, B, C, s);
147        for (int i = 0; i < num; ++i) {
148            if (approximately_zero(s[i])) {
149                return num;
150            }
151        }
152        s[num++] = 0;
153        return num;
154    }
155    if (approximately_zero(A + B + C + D)) {  // 1 is one root
156        int num = SkDQuad::RootsReal(A, A + B, -D, s);
157        for (int i = 0; i < num; ++i) {
158            if (AlmostEqualUlps(s[i], 1)) {
159                return num;
160            }
161        }
162        s[num++] = 1;
163        return num;
164    }
165    double a, b, c;
166    {
167        double invA = 1 / A;
168        a = B * invA;
169        b = C * invA;
170        c = D * invA;
171    }
172    double a2 = a * a;
173    double Q = (a2 - b * 3) / 9;
174    double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
175    double R2 = R * R;
176    double Q3 = Q * Q * Q;
177    double R2MinusQ3 = R2 - Q3;
178    double adiv3 = a / 3;
179    double r;
180    double* roots = s;
181    if (R2MinusQ3 < 0) {   // we have 3 real roots
182        double theta = acos(R / sqrt(Q3));
183        double neg2RootQ = -2 * sqrt(Q);
184
185        r = neg2RootQ * cos(theta / 3) - adiv3;
186        *roots++ = r;
187
188        r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
189        if (!AlmostEqualUlps(s[0], r)) {
190            *roots++ = r;
191        }
192        r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
193        if (!AlmostEqualUlps(s[0], r) && (roots - s == 1 || !AlmostEqualUlps(s[1], r))) {
194            *roots++ = r;
195        }
196    } else {  // we have 1 real root
197        double sqrtR2MinusQ3 = sqrt(R2MinusQ3);
198        double A = fabs(R) + sqrtR2MinusQ3;
199        A = SkDCubeRoot(A);
200        if (R > 0) {
201            A = -A;
202        }
203        if (A != 0) {
204            A += Q / A;
205        }
206        r = A - adiv3;
207        *roots++ = r;
208        if (AlmostEqualUlps(R2, Q3)) {
209            r = -A / 2 - adiv3;
210            if (!AlmostEqualUlps(s[0], r)) {
211                *roots++ = r;
212            }
213        }
214    }
215    return static_cast<int>(roots - s);
216}
217
218// from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf
219// c(t)  = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3
220// c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2
221//       = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2
222static double derivative_at_t(const double* src, double t) {
223    double one_t = 1 - t;
224    double a = src[0];
225    double b = src[2];
226    double c = src[4];
227    double d = src[6];
228    return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t);
229}
230
231// OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t?
232SkDVector SkDCubic::dxdyAtT(double t) const {
233    SkDVector result = { derivative_at_t(&fPts[0].fX, t), derivative_at_t(&fPts[0].fY, t) };
234    return result;
235}
236
237// OPTIMIZE? share code with formulate_F1DotF2
238int SkDCubic::findInflections(double tValues[]) const {
239    double Ax = fPts[1].fX - fPts[0].fX;
240    double Ay = fPts[1].fY - fPts[0].fY;
241    double Bx = fPts[2].fX - 2 * fPts[1].fX + fPts[0].fX;
242    double By = fPts[2].fY - 2 * fPts[1].fY + fPts[0].fY;
243    double Cx = fPts[3].fX + 3 * (fPts[1].fX - fPts[2].fX) - fPts[0].fX;
244    double Cy = fPts[3].fY + 3 * (fPts[1].fY - fPts[2].fY) - fPts[0].fY;
245    return SkDQuad::RootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues);
246}
247
248static void formulate_F1DotF2(const double src[], double coeff[4]) {
249    double a = src[2] - src[0];
250    double b = src[4] - 2 * src[2] + src[0];
251    double c = src[6] + 3 * (src[2] - src[4]) - src[0];
252    coeff[0] = c * c;
253    coeff[1] = 3 * b * c;
254    coeff[2] = 2 * b * b + c * a;
255    coeff[3] = a * b;
256}
257
258/** SkDCubic'(t) = At^2 + Bt + C, where
259    A = 3(-a + 3(b - c) + d)
260    B = 6(a - 2b + c)
261    C = 3(b - a)
262    Solve for t, keeping only those that fit between 0 < t < 1
263*/
264int SkDCubic::FindExtrema(double a, double b, double c, double d, double tValues[2]) {
265    // we divide A,B,C by 3 to simplify
266    double A = d - a + 3*(b - c);
267    double B = 2*(a - b - b + c);
268    double C = b - a;
269
270    return SkDQuad::RootsValidT(A, B, C, tValues);
271}
272
273/*  from SkGeometry.cpp
274    Looking for F' dot F'' == 0
275
276    A = b - a
277    B = c - 2b + a
278    C = d - 3c + 3b - a
279
280    F' = 3Ct^2 + 6Bt + 3A
281    F'' = 6Ct + 6B
282
283    F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
284*/
285int SkDCubic::findMaxCurvature(double tValues[]) const {
286    double coeffX[4], coeffY[4];
287    int i;
288    formulate_F1DotF2(&fPts[0].fX, coeffX);
289    formulate_F1DotF2(&fPts[0].fY, coeffY);
290    for (i = 0; i < 4; i++) {
291        coeffX[i] = coeffX[i] + coeffY[i];
292    }
293    return RootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues);
294}
295
296SkDPoint SkDCubic::top(double startT, double endT) const {
297    SkDCubic sub = subDivide(startT, endT);
298    SkDPoint topPt = sub[0];
299    if (topPt.fY > sub[3].fY || (topPt.fY == sub[3].fY && topPt.fX > sub[3].fX)) {
300        topPt = sub[3];
301    }
302    double extremeTs[2];
303    if (!sub.monotonicInY()) {
304        int roots = FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, sub[3].fY, extremeTs);
305        for (int index = 0; index < roots; ++index) {
306            double t = startT + (endT - startT) * extremeTs[index];
307            SkDPoint mid = ptAtT(t);
308            if (topPt.fY > mid.fY || (topPt.fY == mid.fY && topPt.fX > mid.fX)) {
309                topPt = mid;
310            }
311        }
312    }
313    return topPt;
314}
315
316SkDPoint SkDCubic::ptAtT(double t) const {
317    if (0 == t) {
318        return fPts[0];
319    }
320    if (1 == t) {
321        return fPts[3];
322    }
323    double one_t = 1 - t;
324    double one_t2 = one_t * one_t;
325    double a = one_t2 * one_t;
326    double b = 3 * one_t2 * t;
327    double t2 = t * t;
328    double c = 3 * one_t * t2;
329    double d = t2 * t;
330    SkDPoint result = {a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX + d * fPts[3].fX,
331            a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY + d * fPts[3].fY};
332    return result;
333}
334
335/*
336 Given a cubic c, t1, and t2, find a small cubic segment.
337
338 The new cubic is defined as points A, B, C, and D, where
339 s1 = 1 - t1
340 s2 = 1 - t2
341 A = c[0]*s1*s1*s1 + 3*c[1]*s1*s1*t1 + 3*c[2]*s1*t1*t1 + c[3]*t1*t1*t1
342 D = c[0]*s2*s2*s2 + 3*c[1]*s2*s2*t2 + 3*c[2]*s2*t2*t2 + c[3]*t2*t2*t2
343
344 We don't have B or C. So We define two equations to isolate them.
345 First, compute two reference T values 1/3 and 2/3 from t1 to t2:
346
347 c(at (2*t1 + t2)/3) == E
348 c(at (t1 + 2*t2)/3) == F
349
350 Next, compute where those values must be if we know the values of B and C:
351
352 _12   =  A*2/3 + B*1/3
353 12_   =  A*1/3 + B*2/3
354 _23   =  B*2/3 + C*1/3
355 23_   =  B*1/3 + C*2/3
356 _34   =  C*2/3 + D*1/3
357 34_   =  C*1/3 + D*2/3
358 _123  = (A*2/3 + B*1/3)*2/3 + (B*2/3 + C*1/3)*1/3 = A*4/9 + B*4/9 + C*1/9
359 123_  = (A*1/3 + B*2/3)*1/3 + (B*1/3 + C*2/3)*2/3 = A*1/9 + B*4/9 + C*4/9
360 _234  = (B*2/3 + C*1/3)*2/3 + (C*2/3 + D*1/3)*1/3 = B*4/9 + C*4/9 + D*1/9
361 234_  = (B*1/3 + C*2/3)*1/3 + (C*1/3 + D*2/3)*2/3 = B*1/9 + C*4/9 + D*4/9
362 _1234 = (A*4/9 + B*4/9 + C*1/9)*2/3 + (B*4/9 + C*4/9 + D*1/9)*1/3
363       =  A*8/27 + B*12/27 + C*6/27 + D*1/27
364       =  E
365 1234_ = (A*1/9 + B*4/9 + C*4/9)*1/3 + (B*1/9 + C*4/9 + D*4/9)*2/3
366       =  A*1/27 + B*6/27 + C*12/27 + D*8/27
367       =  F
368 E*27  =  A*8    + B*12   + C*6     + D
369 F*27  =  A      + B*6    + C*12    + D*8
370
371Group the known values on one side:
372
373 M       = E*27 - A*8 - D     = B*12 + C* 6
374 N       = F*27 - A   - D*8   = B* 6 + C*12
375 M*2 - N = B*18
376 N*2 - M = C*18
377 B       = (M*2 - N)/18
378 C       = (N*2 - M)/18
379 */
380
381static double interp_cubic_coords(const double* src, double t) {
382    double ab = SkDInterp(src[0], src[2], t);
383    double bc = SkDInterp(src[2], src[4], t);
384    double cd = SkDInterp(src[4], src[6], t);
385    double abc = SkDInterp(ab, bc, t);
386    double bcd = SkDInterp(bc, cd, t);
387    double abcd = SkDInterp(abc, bcd, t);
388    return abcd;
389}
390
391SkDCubic SkDCubic::subDivide(double t1, double t2) const {
392    if (t1 == 0 || t2 == 1) {
393        if (t1 == 0 && t2 == 1) {
394            return *this;
395        }
396        SkDCubicPair pair = chopAt(t1 == 0 ? t2 : t1);
397        SkDCubic dst = t1 == 0 ? pair.first() : pair.second();
398        return dst;
399    }
400    SkDCubic dst;
401    double ax = dst[0].fX = interp_cubic_coords(&fPts[0].fX, t1);
402    double ay = dst[0].fY = interp_cubic_coords(&fPts[0].fY, t1);
403    double ex = interp_cubic_coords(&fPts[0].fX, (t1*2+t2)/3);
404    double ey = interp_cubic_coords(&fPts[0].fY, (t1*2+t2)/3);
405    double fx = interp_cubic_coords(&fPts[0].fX, (t1+t2*2)/3);
406    double fy = interp_cubic_coords(&fPts[0].fY, (t1+t2*2)/3);
407    double dx = dst[3].fX = interp_cubic_coords(&fPts[0].fX, t2);
408    double dy = dst[3].fY = interp_cubic_coords(&fPts[0].fY, t2);
409    double mx = ex * 27 - ax * 8 - dx;
410    double my = ey * 27 - ay * 8 - dy;
411    double nx = fx * 27 - ax - dx * 8;
412    double ny = fy * 27 - ay - dy * 8;
413    /* bx = */ dst[1].fX = (mx * 2 - nx) / 18;
414    /* by = */ dst[1].fY = (my * 2 - ny) / 18;
415    /* cx = */ dst[2].fX = (nx * 2 - mx) / 18;
416    /* cy = */ dst[2].fY = (ny * 2 - my) / 18;
417    // FIXME: call align() ?
418    return dst;
419}
420
421void SkDCubic::align(int endIndex, int ctrlIndex, SkDPoint* dstPt) const {
422    if (fPts[endIndex].fX == fPts[ctrlIndex].fX) {
423        dstPt->fX = fPts[endIndex].fX;
424    }
425    if (fPts[endIndex].fY == fPts[ctrlIndex].fY) {
426        dstPt->fY = fPts[endIndex].fY;
427    }
428}
429
430void SkDCubic::subDivide(const SkDPoint& a, const SkDPoint& d,
431                         double t1, double t2, SkDPoint dst[2]) const {
432    SkASSERT(t1 != t2);
433#if 0
434    double ex = interp_cubic_coords(&fPts[0].fX, (t1 * 2 + t2) / 3);
435    double ey = interp_cubic_coords(&fPts[0].fY, (t1 * 2 + t2) / 3);
436    double fx = interp_cubic_coords(&fPts[0].fX, (t1 + t2 * 2) / 3);
437    double fy = interp_cubic_coords(&fPts[0].fY, (t1 + t2 * 2) / 3);
438    double mx = ex * 27 - a.fX * 8 - d.fX;
439    double my = ey * 27 - a.fY * 8 - d.fY;
440    double nx = fx * 27 - a.fX - d.fX * 8;
441    double ny = fy * 27 - a.fY - d.fY * 8;
442    /* bx = */ dst[0].fX = (mx * 2 - nx) / 18;
443    /* by = */ dst[0].fY = (my * 2 - ny) / 18;
444    /* cx = */ dst[1].fX = (nx * 2 - mx) / 18;
445    /* cy = */ dst[1].fY = (ny * 2 - my) / 18;
446#else
447    // this approach assumes that the control points computed directly are accurate enough
448    SkDCubic sub = subDivide(t1, t2);
449    dst[0] = sub[1] + (a - sub[0]);
450    dst[1] = sub[2] + (d - sub[3]);
451#endif
452    if (t1 == 0 || t2 == 0) {
453        align(0, 1, t1 == 0 ? &dst[0] : &dst[1]);
454    }
455    if (t1 == 1 || t2 == 1) {
456        align(3, 2, t1 == 1 ? &dst[0] : &dst[1]);
457    }
458    if (precisely_subdivide_equal(dst[0].fX, a.fX)) {
459        dst[0].fX = a.fX;
460    }
461    if (precisely_subdivide_equal(dst[0].fY, a.fY)) {
462        dst[0].fY = a.fY;
463    }
464    if (precisely_subdivide_equal(dst[1].fX, d.fX)) {
465        dst[1].fX = d.fX;
466    }
467    if (precisely_subdivide_equal(dst[1].fY, d.fY)) {
468        dst[1].fY = d.fY;
469    }
470}
471
472/* classic one t subdivision */
473static void interp_cubic_coords(const double* src, double* dst, double t) {
474    double ab = SkDInterp(src[0], src[2], t);
475    double bc = SkDInterp(src[2], src[4], t);
476    double cd = SkDInterp(src[4], src[6], t);
477    double abc = SkDInterp(ab, bc, t);
478    double bcd = SkDInterp(bc, cd, t);
479    double abcd = SkDInterp(abc, bcd, t);
480
481    dst[0] = src[0];
482    dst[2] = ab;
483    dst[4] = abc;
484    dst[6] = abcd;
485    dst[8] = bcd;
486    dst[10] = cd;
487    dst[12] = src[6];
488}
489
490SkDCubicPair SkDCubic::chopAt(double t) const {
491    SkDCubicPair dst;
492    if (t == 0.5) {
493        dst.pts[0] = fPts[0];
494        dst.pts[1].fX = (fPts[0].fX + fPts[1].fX) / 2;
495        dst.pts[1].fY = (fPts[0].fY + fPts[1].fY) / 2;
496        dst.pts[2].fX = (fPts[0].fX + 2 * fPts[1].fX + fPts[2].fX) / 4;
497        dst.pts[2].fY = (fPts[0].fY + 2 * fPts[1].fY + fPts[2].fY) / 4;
498        dst.pts[3].fX = (fPts[0].fX + 3 * (fPts[1].fX + fPts[2].fX) + fPts[3].fX) / 8;
499        dst.pts[3].fY = (fPts[0].fY + 3 * (fPts[1].fY + fPts[2].fY) + fPts[3].fY) / 8;
500        dst.pts[4].fX = (fPts[1].fX + 2 * fPts[2].fX + fPts[3].fX) / 4;
501        dst.pts[4].fY = (fPts[1].fY + 2 * fPts[2].fY + fPts[3].fY) / 4;
502        dst.pts[5].fX = (fPts[2].fX + fPts[3].fX) / 2;
503        dst.pts[5].fY = (fPts[2].fY + fPts[3].fY) / 2;
504        dst.pts[6] = fPts[3];
505        return dst;
506    }
507    interp_cubic_coords(&fPts[0].fX, &dst.pts[0].fX, t);
508    interp_cubic_coords(&fPts[0].fY, &dst.pts[0].fY, t);
509    return dst;
510}
511